Array for storing coefficients of a polynomial of three variables with specified degree. More...

#include <polynomial.h>

Public Member Functions
	PolynomialCoeffs ()=delete

	PolynomialCoeffs (const double constant, const double x, const double y, const double z)
	Create a linear function with the specified coefficients.

	PolynomialCoeffs (const unsigned N)
	Create a polynomial of degree N.

unsigned	get_degree () const
	Get the degree of the polynomial.

double	operator() (const std::array< double, 3 > &pt) const
	Evaluate the polynomial at the specified point.

PolynomialCoeffs	operator* (const double c) const
	Multiply a polynomial by a constant value.

PolynomialCoeffs	operator* (const PolynomialCoeffs &rhs) const
	Multiply two polynomials together.

PolynomialCoeffs &	operator*= (const double c)
	Scale the polynomial by a constant.

PolynomialCoeffs	operator+ (const PolynomialCoeffs &rhs) const
	Add two polynomials together.

PolynomialCoeffs &	operator+= (const PolynomialCoeffs &rhs)
	Adds another polynomial to this one.

double &	operator[] (const std::array< unsigned, 3 > &pows)
	Access the coefficient for the specified term.

double	operator[] (const std::array< unsigned, 3 > &pows) const
	Access the coefficient for the specified term.

PolynomialCoeffs	pow (const unsigned j) const
	Expands a linear polynomial raised to a power.

Static Protected Member Functions
static unsigned	polyIndex (const std::array< unsigned, 3 > &pows)
	Gets the array index for the coefficient of a particular term.

Protected Attributes
std::vector< double >	coeffs
	Array storing the coefficients.

unsigned	degree
	The degree of the polynomial.

Detailed Description

Array for storing coefficients of a polynomial of three variables with specified degree.

The general polynomial of three variables is written as

$\sum_{i,j,k} c_{i,j,k} x^i y^j z^k,$

where $0\le i+j+k\le N$ . This data structure stores the coefficients in a vector and provides access routines in terms of the exponents $i$ , $j$ , and $k$ .

Note: This class is not designed for polynomials with a large degree.

Constructor & Destructor Documentation

◆ PolynomialCoeffs() [1/3]

slymer::PolynomialCoeffs::PolynomialCoeffs ( )

delete

◆ PolynomialCoeffs() [2/3]

slymer::PolynomialCoeffs::PolynomialCoeffs	(	const double	constant,
		const double	x,
		const double	y,
		const double	z
	)

inline

Create a linear function with the specified coefficients.

Parameters

[in]	constant	The constant term.
[in]	x	The coefficient on x.
[in]	y	The coefficient on y.
[in]	z	The coefficient on z.

References coeffs, constant(), and polyIndex().

◆ PolynomialCoeffs() [3/3]

slymer::PolynomialCoeffs::PolynomialCoeffs ( const unsigned N )

inline

Create a polynomial of degree N.

Parameters

[in] N The degree of the polynomial.

References coeffs.

Member Function Documentation

◆ get_degree()

unsigned slymer::PolynomialCoeffs::get_degree ( ) const

inline

Get the degree of the polynomial.

Returns: The degree of the polynomial.

References degree.

Referenced by slymer::PrimitiveGaussian::PrimitiveGaussian().

◆ operator()()

double slymer::PolynomialCoeffs::operator() ( const std::array< double, 3 > & pt ) const

Evaluate the polynomial at the specified point.

Note: This implemention is a bit naive; it is not intended for polynomials with a large degree.

Parameters

[in] pt The point at which the polynomial is evaluated.

Returns: The value of the polynomial at the point.

References coeffs, degree, k, polyIndex(), and pow().

◆ operator*() [1/2]

PolynomialCoeffs slymer::PolynomialCoeffs::operator* ( const double c ) const

Multiply a polynomial by a constant value.

*this is one of the polynomials.

Parameters

[in] c The constant value.

Returns: The product of c and the polynomial.

References c, and coeffs.

◆ operator*() [2/2]

PolynomialCoeffs slymer::PolynomialCoeffs::operator* ( const PolynomialCoeffs & rhs ) const

Multiply two polynomials together.

*this is one of the polynomials.

Parameters

[in] rhs The other polynomial.

Returns: The product of *this and rhs.

References degree, and operator[]().

◆ operator*=()

PolynomialCoeffs & slymer::PolynomialCoeffs::operator*= ( const double c )

Scale the polynomial by a constant.

Parameters

[in] c The scale factor.

Returns: The scaled polynomial (*this).

References c, and coeffs.

◆ operator+()

PolynomialCoeffs slymer::PolynomialCoeffs::operator+ ( const PolynomialCoeffs & rhs ) const

Add two polynomials together.

*this is one of the polynomials.

Parameters

[in] rhs The other polynomial.

Returns: The sum of *this and rhs.

References coeffs, and degree.

◆ operator+=()

PolynomialCoeffs & slymer::PolynomialCoeffs::operator+= ( const PolynomialCoeffs & rhs )

Adds another polynomial to this one.

Parameters

[in] rhs The other polynomial.

Returns: The sum of *this and rhs, in *this.

References coeffs, and degree.

◆ operator[]() [1/2]

double & slymer::PolynomialCoeffs::operator[] ( const std::array< unsigned, 3 > & pows )

inline

Access the coefficient for the specified term.

Parameters

[in] pows The powers of x, y, and z for this term.

Returns: Reference to the array term.

References coeffs, and polyIndex().

Referenced by operator*().

◆ operator[]() [2/2]

double slymer::PolynomialCoeffs::operator[] ( const std::array< unsigned, 3 > & pows ) const

inline

Access the coefficient for the specified term.

Parameters

[in] pows The powers of x, y, and z for this term.

Returns: The array term.

References coeffs, and polyIndex().

◆ polyIndex()

static unsigned slymer::PolynomialCoeffs::polyIndex ( const std::array< unsigned, 3 > & pows )

inlinestaticprotected

Gets the array index for the coefficient of a particular term.

Terms of degree 0 are stored before those of degree 1, etc.

Within a particular degree, think of the problem in the combinatorial "stars and bars" idea: there are n stars and two bars dividing them into three parts. (Each part corresponds to the number of x, y, or z factors.) The first array element will be bars in the 0 and 1 positions, then 0 and 2, ..., 0 and n+1, then 1 and 2, etc.

Parameters

[in] pows The powers of x, y, and z for this term.

Returns: The array offset, as described above.

Referenced by PolynomialCoeffs(), operator()(), operator[](), and operator[]().

◆ pow()

PolynomialCoeffs slymer::PolynomialCoeffs::pow ( const unsigned j ) const

Expands a linear polynomial raised to a power.

Essentially expands the polynomial

$(coeffs[0] + coeffs[1]*x + coeffs[2]*y + coeffs[3]*z)^j$

into a new PolynomialCoeffs object. That is, output[{{p,q,r}}] is the coefficient on the $x^p y^q z^r$ term.

This function requires multinomial coefficients, which involve factorials. We assume that the arguments to the factorials will not be large and hard-code small values. An error will be thrown if the maximum value is exceeded.

Exceptions

std::runtime_error if the polynomial does not have degree 1.

Parameters

[in] j The exponent of the expansion.

Returns: The polynomial, ordered using the polyIndex scheme.

References degree, and pow().

Referenced by operator()(), and pow().

Member Data Documentation

◆ coeffs

std::vector<double> slymer::PolynomialCoeffs::coeffs

protected

Array storing the coefficients.

For a given degree n, there are $(n+1)(n+2)/2$ monomials. The total array thus has $(n+1)*(n+2)*(n+3)/6$ terms. The array stores all terms with degree 0, then degree 1, then degree 2, etc.